The present invention relates to digital imaging.
An image can be modeled as a piecewise continuous two dimensional function representing color values over a rectangular plane (the image plane). A raster image (suitable for storage/retrieval processing in digital computers) is a discrete two dimensional array of image color values. A raster image can be obtained by sampling a continuous image function over a regular grid spanning the image plane. Image resampling is the process of mapping one raster image to another, corresponding to the same underlying image, but defined over a different sampling grid. Image magnification (also referred to as image supersampling) is a special case of image resampling. In this case, an input raster image is mapped to another raster image (corresponding to the same underlying image) on a denser sampling grid of larger size. Using conventional methods, magnification of raster images typically leads to edge blurring, a phenomenon whereby sharp changes of color in the original raster image get replaced by gradual color variation in the output raster image. Since, in a typical image, edges demarcate visually meaningful segments of the image, edge blurring degrades the visual quality of the image.
In a typical digital imaging system, any original underlying piecewise continuous image function is lost, the raster image is all that is remembered. Consequently, during resampling, when it is necessary to generate color values at locations other than original sampled points, some estimating technique is needed. Such techniques implicitly or explicit solve the problem of image reconstruction, whereby they make a local or global estimation of the lost continuous image function which can then be resampled as desired.
Assuming an original continuous image was sampled at rate satisfying the Nyquist criterion, it is possible, in theory, to have an ideal reconstruction filter. The ideal reconstruction filter is shaped like a box in the frequency domain (equivalently, it is the sinc function in spatial domain). The width of the box is determined by ƒNyquist=2ƒmax, where ƒmax is the largest spatial frequency component in the image. Unfortunately, in typical cases, ƒmax is not known a priori. An ideal reconstruction filter is also undesirable from a performance point of view, because the sinc function converges very slowly with its magnitude going down very slowly over time (Gibbs phenomenon).
In the absence of the ideal reconstruction filter, most image reconstruction and resampling techniques fall back to reconstruction filters whose characteristics are similar to the ideal one. Among nonideal reconstruction filters, interpolators enjoy popularity for performance and simplicity reasons. Interpolation is the process of determining the values of a discrete function at points lying in between samples. This is achieved by fitting a continuous function through the discrete input samples (note the implicit regularization assumption that the unknown function is continuous in between samples—this directly leads to edge blurring). From a signal processing point of view, interpolation is equivalent to applying a low-pass FIR (Finite Impulse Response) filter whose characteristics resemble (to varying extents) those of the ideal reconstruction filter. Local interpolation techniques do not enforce any global smoothness measure. A local polynomial surface patch is fitted to the immediate neighborhood of the point in question without any attempt to ensure inter-patch continuity.